#719406

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(9,10,13,16)(11,14,15,12)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31);(5,6,7,8)(9,11,13,15)(10,14,16,12)(18,22)(23,30)(24,27)(31,32);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31);(1,2,3,4)(9,12,13,14)(10,15,16,11)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32);(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 626602);